24th International Colloquium Tribology - January 2024 171 Towards a Continuum Description of Lubrication in Highly Pressurized Nanometer-wide Constrictions: the Importance of Accurate Slip Laws Andrea Codrignani 1,2 , Stefan Peeters 1 , Hannes Holey 1,2 , Franziska Stief 1,3 , Daniele Savio 1,4 , Lars Pastewka 5 , Gianpietro Moras 1 , Kerstin Falk 1 and Michael Moseler 1,2,3,* 1 Microtribology Center μTC, Fraunhofer Institute for Mechanics of Materials IWM, Wöhlerstraße 11, 79108 Freiburg, Germany 2 Freiburg Materials Research Center, University of Freiburg, Stefan-Meier-Straße 21, 79104 Freiburg, Germany. 3 Institute of Physics, University of Freiburg, Hermann-Herder-Straße 3a, 79104 Freiburg, Germany. 4 Freudenberg Technology Innovation SE & Co.KG, Höhnerweg 2-4, 69469 Weinheim, Germany 5 Department of Microsystems Engineering, University of Freiburg, Georges-Köhler-Allee 103, 79110 Freiburg, Germany * michael.moseler@iwm.fraunhofer.de 1. Introduction Modern, compact and efficient tribological systems are often operated in mixed lubrication or even boundary lubrication, meaning small gaps, frequently high pressures and occasional solid-solid contacts between the lubricated sliding surfaces (1). In particular, the need for climate-friendly lubricants with low viscosities (2), an increase in assembly precision of lubricated contacts for electric vehicles (3) and new high-performance coating techniques (4) that allow smaller assembly tolerances have driven the shift to operating devices in the mixed lubrication regime. Downsizing dramatically increases loads in tribological components (often in the GPa range) resulting in an additional driving force towards boundary lubricated contacts (5). Under such severe conditions, the film thickness in typical applications can reach a few nanometers (6), becoming comparable to the size of the lubricant molecules themselves. At this scale, a current state-of-the-art continuum description of lubricant flow is expected to lose its validity due to density layering (7), solvation forces (8), the emergence of solid-like states (9, 10), increased viscosities or wall slip (11-16) - see also (17) for a comprehensive review of atomistic simulations of confined lubricant films. The Reynolds Lubrication Equation (RLE) is the most commonly used continuum equation for flow calculations in lubricated systems (18). Although the RLE was proposed at the end of the 19 th century for incompressible laminar Newtonian flows (19), the past decades have seen research in extending its applicability to lubricants exhibiting compressibility, piezoviscosity, shear thinning and cavitation (1, 11, 20). These extensions have rendered the RLE a predictive description for elastohydrodynamic lubrication (EHL) of technically relevant tribocontacts (20), provided quantitative constitutive laws for compressibility and viscosity are employed (21). In combination with empirical friction coefficients for boundary lubrication regions, the RLE is also used for mixed lubrication problems (22, 23). Technically, the RLE is employed for local gap heights exceeding a certain threshold (of the order of 0.1 - 1 µm) to obtain the hydrodynamic contribution to friction while smaller gaps are assumed to be in solid-solid contact and modeled via a Coulomb-Amontons friction law or a Bowden-Tabor (24) constant interfacial shear stress. The choice of this threshold is more a matter of convenience than of a physical reasoning. It would be very useful to explore the lower gap size limits for a continuum description of the lubricant flow - especially for high local pressures characteristic for EHL contacts. By extending the RLE realm to smaller scales the importance of empirical solid-solid contact friction laws could be reduced and therefore the predictive power of mixed lubrication calculations would improve substantially. In the present work, isothermal non-equilibrium molecular dynamics (MD) simulations of hexadecane in a gold converging-diverging channel (depicted in Fig. 1A and B) are performed to generate realistic benchmark data representative of mineral oils lubricating an asperity contact between two metal surfaces. In a previous related study, some of the authors have used this alkane/ gold model to study the onset of cavitation and its continuum description in a parallel channel with heterogeneous slip conditions at moderate pressures (25), while the current work addresses much higher pressures and a variation in the channel height (26). A profound MD characterization of our atomistic hexadecane lubricant model provides an equation of state ρ( p) as well as a pressureand shear-rate-dependent constitutive law for the viscosity h( p, γ ̇). With these data the validity limit of the RLE description for gaps h 0 in the single-digit nanometer range and pressures p approaching the GPa regime is explored. A failure of such a traditional RLE treatment for high pressures can be traced back to the violation of the no-slip boundary condition on the gold (111) facets in the converging-diverging channel. By a separate parametric MD study of hexadecane in parallel gold channels (Fig. 1C and D) the pressure-dependence of wall slip is quantified and the existence of a constitutive law v s- =-v s (τ, p) that relates local shear stress τ at the wall with slip velocity v s is demonstrated. A v s (τ, p) law is also found for technically relevant systems such as diamond-like carbon (DLC) channels filled with poly-α-olefin (PAO) lubricant (Fig. 1E). Finally, we show that an extension of the RLE by v s- =-v s (τ, p) results in a model that allows for a quantitative description of pressure and velocity profiles in our MD simulations of converging-diverging channels for minimum gap sizes and local pressure that are of the order of 1 nm and 1 GPa, respectively. 172 24th International Colloquium Tribology - January 2024 Towards a Continuum Description of Lubrication in Highly Pressurized Nanometer-wide Constrictions: the Importance of Accurate Slip Laws Fig. 1: Atomistic models used in the molecular dynamics simulations. References [1] Y. Meng, J. Xu, Z. Jin, B. Prakash, Y. Hu, A review of recent advances in tribology. Friction 8, 221-300 (2020). [2] S.-W. Zhang, Recent developments of green tribology. Surface Topography: Metrology and Properties 4, 23004 (2016). [3] L. I. 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